Printing transferable components using microstructured elastomeric surfaces with pressure modulated reversible adhesion

ABSTRACT

In a method of printing a transferable component, a stamp including an elastomeric post having three-dimensional relief features protruding from a surface thereof is pressed against a component on a donor substrate with a first pressure that is sufficient to mechanically deform the relief features and a region of the post between the relief features to contact the component over a first contact area. The stamp is retracted from the donor substrate such that the component is adhered to the stamp. The stamp including the component adhered thereto is pressed against a receiving substrate with a second pressure that is less than the first pressure to contact the component over a second contact area that is smaller than the first contact area. The stamp is then retracted from the receiving substrate to delaminate the component from the stamp and print the component onto the receiving substrate. Related apparatus and stamps are also discussed.

RELATED APPLICATIONS

This application is a divisional of U.S. patent application Ser. No.13/237,375 filed Sep. 20, 2011 which is related to U.S. patentapplication Ser. No. 11/423,192 filed Jun. 9, 2006, and U.S. patentapplication Publication Ser. No. 12/621,804, filed Nov. 19, 2009, thedisclosures of which are specifically incorporated by reference to theextent not inconsistent herewith.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made, at least in part, with United Statesgovernmental support by a National Security Science and EngineeringFaculty Fellowship and the U.S. Department of Energy, Division ofMaterials Sciences under Award No. DEFG02-91ER45439. The United StatesGovernment has certain rights in this invention.

BACKGROUND OF THE INVENTION

Semiconductor chip or die automated assembly equipments typically relyon the use of vacuum operated placement heads often referred to asvacuum grippers or pick-and-place tools. In their simplest embodiment,these placement heads typically consist of an open ended cylinder havinga drilled nozzle surface which seals to the die to accomplish physicalattachment. Semiconductor chips or die which are ultra thin, fragile, ortoo small cannot be economically handled by conventional vacuumgrippers. As a result, alternative approaches such as self-assembly ordry transfer printing technologies are being investigated.

Transfer printing enables the massively parallel assembly of highperformance semiconductor devices onto virtually any substrate material,including glass, plastics, metals or other semiconductors (see, e.g.,U.S. patent application Ser. No. 11/145,574 entitled “METHODS ANDDEVICES FOR FABRICATING AND ASSEMBLING PRINTABLE SEMICONDUCTORELEMENTS,” filed Jun. 2, 2005). This semiconductor transfer printingtechnology relies on the use of a microstructured elastomeric stamp toselectively pick-up devices from a source wafer, and then prints thesedevices onto a target substrate. The transfer process is massivelyparallel as the stamps are designed to transfer hundreds to thousands ofdiscrete structures in a single pick-up and print operation.

While pick-and-place tools rely on suction forces, dry transfer printingtools rely on surface adhesion forces to control the pickup and releaseof the semiconductor devices. To enable dry transfer printing, methodsto control the adhesion forces between the semiconductor elements andthe elastomeric stamp are required. One such method is described in U.S.patent application Ser. No. 11/423,192 filed Jun. 9, 2006, entitled“PATTERN TRANSFER PRINTING BY KINETIC CONTROL OF ADHESION TO ANELASTOMERIC STAMP.” In that method, the elastomeric stamp adhesionforces are controlled by adjusting the delamination rate of theelastomeric transfer stamp. This control of separation or delaminationrate provides a means of increasing the stamp adhesion forces that maybe used to pickup semiconductor elements from a source wafer. There maybe problems, however, associated with transferring the semiconductorelements from the stamp to a receiving substrate with this technique.For example, stamps optimized for dry transfer printing semiconductorelements with high placement accuracy typically use a stiff backinglayer. During the printing or transfer step, the delamination rate ofthose stamps can be unstable and/or difficult to control when the stiffbacking layer(s) are subject to bending forces. Also, printing yields onsurfaces that are not ultra smooth, and/or on low tack surfaces, can bevery low.

Accordingly, there is a need for improved methods for transfer printingsemiconductor elements.

SUMMARY OF THE INVENTION

It should be appreciated that this Summary is provided to introduce aselection of concepts in a simplified form, the concepts being furtherdescribed below in the Detailed Description. This Summary is notintended to identify key features or essential features of thisdisclosure, nor is it intended to limit the scope of the disclosure.

Provided are methods and systems for printing transferable components.In methods according to some embodiments of the present invention, astamp including an elastomeric post having three-dimensional relieffeatures protruding from a surface thereof is pressed against atransferable component on a donor substrate with a first pressuresufficient to mechanically deform a region of the post between therelief features to contact the component over a first contact area. Thestamp is retraced from the donor substrate such that the component isadhered to the stamp. The stamp including the component adhered theretois pressed against a receiving substrate with a second pressure that isless than the first pressure to contact the component over a secondcontact area that is smaller than the first contact area. The stamp isretracted from the receiving substrate to delaminate the component fromthe stamp and print the component onto the receiving substrate.

In some embodiments, the second pressure is insufficient to mechanicallydeform the region of the post between the relief features to contact thecomponent.

In some embodiments, pressing the stamp with the first pressure issufficient to compress the relief features and collapse the region ofthe post therebetween to contact the component to define the firstcontact area. The first contact area may be substantially similar to across-sectional area of the post taken along a plane parallel to thesurface thereof.

In some embodiments, retracting the stamp from the donor substrateincludes removing the first pressure from the stamp to restore therelief features and the region of the post therebetween such that thecomponent is adhered to the stamp by ends of one or more of the relieffeatures to define the second contact area that is smaller than thefirst contact area.

In some embodiments, an elastic restoring force of the post responsiveto removal of the first pressure is insufficient to delaminate thecomponent from the stamp.

In some embodiments, an adhesive strength provided by the first contactarea is greater than that of the second contact area by a factor of 1000or more.

In some embodiments, pressing the stamp with the second pressure issufficient to compress the relief features without collapse of theregion of the post therebetween such that the relief features contactthe component over a third contact area that is smaller than the firstcontact area but is larger than the second contact area.

In some embodiments, retracting the stamp from the donor substrate isperformed at a first speed to adhere the component thereto, andretracting the stamp from the receiving substrate is performed at asecond speed that is less than the first speed to delaminate thecomponent therefrom. The first speed is sufficient to fracture aninterface between the component and the donor substrate withoutfracturing an interface defined by the first contact area between thestamp and the component. A combination of the first contact area definedby the first pressure, a viscoelastic property of the elastomeric post,and the first speed of retracting may be sufficient to adhere thecomponent to the stamp.

In some embodiments, the first speed may be about 5 micrometers persecond or more, and the second speed may be about 1 millimeter persecond or less.

In some embodiments, the relief features are positioned around aperiphery of the surface of the post.

In some embodiments, the plurality of relief features are first relieffeatures, and the surface of the post further includes a secondthree-dimensional relief feature that protrudes from the region thereofbetween the first relief features. The second relief feature may belarger than the first relief features in at least one dimension.

In some embodiments, the elastomeric post and the relief featuresprotruding from the surface thereof comprise polydimethylsiloxane(PDMS). Respective ends of the relief features may have pyramidal,conical, and/or spherical geometries.

An apparatus for printing transferable components according to someembodiments of the present invention includes a stamp, a transferprinting tool head including the stamp mounted thereon, and a controllerconfigured to operate the transfer printing tool head. The stampincludes at least one elastomeric post protruding therefrom, where thepost has a surface configured for contact with a respective transferablecomponent and includes three-dimensional relief features protrudingtherefrom. The controller is configured to operate the transfer printingtool head to contact the stamp including the post protruding therefromwith the respective transferable component on a donor substrate at afirst pressure sufficient to mechanically deform a region of the postbetween the relief features to contact the respective transferablecomponent over a first contact area, retract the stamp from the donorsubstrate such that the respective transferable component is adhered tothe stamp, contact the stamp including the respective transferablecomponent adhered thereto with a receiving substrate at a secondpressure that is less than the first pressure to contact the componentover a second contact area that is smaller than the first contact area,and retract the stamp from the receiving substrate to delaminate therespective transferable component from the stamp and print therespective transferable component onto the receiving substrate.

In some embodiments, the second pressure is insufficient to mechanicallydeform the region of the post between the relief features to contact therespective transferable component.

In some embodiments, the first pressure is sufficient to compress therelief features and collapse the region of the post therebetween tocontact the respective transferable component over a first contact area.The first contact area may be substantially similar to a cross-sectionalarea of the post taken along a plane parallel to the surface thereof.

In some embodiments, the controller is configured to operate thetransfer printing tool head to remove the first pressure from the stampduring retraction from the donor substrate to uncompress the relieffeatures and uncollapse the region of the post therebetween such thatthe respective transferable component is adhered to the stamp by ends ofone or more of the relief features over a second contact area that issmaller than the first contact area.

In some embodiments, the elastomeric post and the relief featuresprotruding from the surface thereof are formed from polydimethylsiloxane(PDMS), and an adhesive strength provided by the first contact area isgreater than that of the second contact area by three or more orders ofmagnitude.

In some embodiments, the controller is configured to operate thetransfer printing tool head to retract the stamp from the donorsubstrate at a first speed to adhere the respective transferablecomponent thereto, and to retract the stamp from the receiving substrateat a second speed that is less than the first speed to delaminate therespective transferable component therefrom.

In some embodiments, the relief features are positioned around aperiphery of the surface of the post. The relief features may be firstrelief features, and the surface of the post may further include asecond three-dimensional relief feature that protrudes from the regionthereof between the first relief features. The second relief feature maybe larger than the first relief features in at least one dimension.

An elastomeric stamp according to some embodiments of the presentinvention includes a deformable elastomeric layer having a postprotruding therefrom. The post includes an elastomeric surfaceconfigured for contact with a transferable component. The surfaceincludes a plurality of three-dimensional relief features protrudingtherefrom, and one or more of the plurality of relief features isconfigured to support the transferable component.

In some embodiments, ones of the plurality of relief features are spacedapart from one another on the surface at a distance sufficient such thata region of the surface therebetween is deformable to contact thetransferable component responsive to application of a predeterminedforce to the stamp.

In some embodiments, the predetermined force is sufficient to compressthe plurality of relief features and collapse the region of the surfacetherebetween to contact the transferable component over a first contactarea that is substantially similar to a cross-sectional area of the posttaken along a plane parallel to the surface thereof. The predeterminedforce may be about 0.39 mN or more.

In some embodiments, ones of the plurality of relief features arepositioned around a periphery of the surface of the post. For example,each of the relief features may be positioned at a respective corner ofthe post

In some embodiments, the plurality of relief features are first relieffeatures, and the surface of the post further includes a secondthree-dimensional relief feature that protrudes from the region thereofbetween the first relief features and is configured to support thetransferable component. The second relief feature may be larger than thefirst relief features in at least one dimension.

In some embodiments, a height of each of the plurality of relieffeatures is based on the distance between the ones of the plurality ofrelief features and an elasticity of the deformable elastomeric layer.For example, the elastomeric layer including the post and the pluralityof relief features may be formed of polydimethylsiloxane (PDMS).

In some embodiments, the height of each of the plurality of relieffeatures is between about 8.5 micrometers and about 13 micrometers.

In some embodiments, respective tips of the relief features may havepyramidal, conical, and/or spherical geometries.

Other methods and/or devices according to some embodiments will becomeapparent to one with skill in the art upon review of the followingdrawings and detailed description. It is intended that all suchadditional embodiments, in addition to any and all combinations of theabove embodiments, be included within this description, be within thescope of the invention, and be protected by the accompanying claims.

Without wishing to be bound by any particular theory, there can bediscussion herein of beliefs or understandings of underlying principlesor mechanisms relating to embodiments of the invention. It is recognizedthat embodiments of the invention can nonetheless be operative anduseful regardless of the ultimate correctness of any explanation orhypothesis presented herein.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A-1F are perspective views of a process flow diagram illustratingmethods of printing transferable components according to someembodiments of the present invention.

FIG. 1G provides perspective views illustrating the stamp of FIGS. 1A-1Fin greater detail.

FIG. 1H provides perspective views illustrating operations for thefabricating a stamp including an elastomeric microstructured transfersurface according to some embodiments of the present invention, and FIG.1I is a schematic illustration of a process for fabricating siliconplatelets in printable configurations by removing a buried oxide layerfrom a silicon-on-insulator (SOI) wafer according to some embodiments ofthe present invention.

FIGS. 2A-2C are scanning electron microscope (SEM) images of a poststructure having a four-tipped layout, as used in the embodiment ofFIGS. 1A-1F.

FIG. 2D is a cross-sectional schematic illustration of the poststructure of FIGS. 1A-1F.

FIGS. 2E and 2F are SEM images showing a post structure having afive-tipped layout according to further embodiments of the presentinvention.

FIGS. 2G and 2H provide side and overhead views of operations andapparatus for transfer printing from a donor substrate to a receiversubstrate using the five-tipped stamp of FIGS. 2E-2F.

FIGS. 3A and 3B are graphs illustrating force-time and force-distancecurves associated with contact of a microtip surface with a siliconwafer in transfer printing operations according to some embodiments ofthe present invention.

FIGS. 3C and 3D are graphs illustrating force-speed plots for removing apost having a microtip surface according to some embodiments of thepresent invention from a silicon wafer as compared to a post having aflat surface, FIG. 3E illustrates an example of a custom measurementsystem according to some embodiments of the present invention, and FIG.3F is a graph illustrating pull-off force data for a stamp with afour-tipped layout in accordance with some embodiments of the presentinvention as compared to as stamp having a flat surface.

FIGS. 4A-4F are SEM images illustrating example transfer printingresults achieved with microstructured elastomeric surfaces and methodsin accordance with some embodiments of the present invention.

FIG. 5 is a schematic illustration of a device capable of transferprinting semiconductor elements according to some embodiments of thepresent invention.

FIG. 6 is a flowchart illustrating example operations for transferprinting using stamps with microstructured elastomeric surfaces inaccordance with embodiments of the present invention.

FIGS. 7A-7C illustrate an example transistor fabricating using transferprinting methods and apparatus according to some embodiments of thepresent invention.

FIGS. 8A-8D are graphs illustrating characteristics of microstructuredelastomeric surfaces according to some embodiments of the presentinvention during transfer printing operations.

FIGS. 9A and 9B are graphs illustrating forces acting on microstructuredelastomeric surfaces according to some embodiments of the presentinvention during transfer printing operations.

FIGS. 10A and 10B are graphs illustrating load vs. displacement formicrostructured elastomeric surfaces in accordance with embodiments ofthe present invention.

DETAILED DESCRIPTION OF EMBODIMENTS OF THE INVENTION

The present invention now will be described more fully hereinafter withreference to the accompanying drawings, in which embodiments of theinvention are shown. However, this invention should not be construed aslimited to the embodiments set forth herein. Rather, these embodimentsare provided so that this disclosure will be thorough and complete, andwill fully convey the scope of the invention to those skilled in theart. In the drawings, the thickness of layers and regions areexaggerated for clarity. Like numbers refer to like elements throughout.

Unless otherwise defined, all terms used in disclosing embodiments ofthe invention, including technical and scientific terms, have the samemeaning as commonly understood by one of ordinary skill in the art towhich this invention belongs, and are not necessarily limited to thespecific definitions known at the time of the present invention beingdescribed. Accordingly, these terms can include equivalent terms thatare created after such time. It will be further understood that terms,such as those defined in commonly used dictionaries, should beinterpreted as having a meaning that is consistent with their meaning inthe present specification and in the context of the relevant art andwill not be interpreted in an idealized or overly formal sense unlessexpressly so defined herein. All publications, patent applications,patents, and other references mentioned herein are incorporated byreference in their entireties.

“Printing” refers to a process of transferring a feature, such as asemiconductor component or element, from a first surface to a secondsurface. In some embodiments, the first surface is a donor surface andthe second surface a receiving surface, and the transfer is mediated byan intermediate surface, such as a stamp having a microstructuredelastomeric transfer surface, which is capable of releasing the elementsto a receiving surface on a target substrate, thereby transferring thesemiconductor element. In some embodiments, the printing is dry transferprinting of printable semiconductors, wherein the adhesive force betweena solid object and the stamp surface is rate-sensitive.

“Stamp” refers to a component for transfer, assembly and/or integrationof structures and materials via printing, for example dry transfercontact printing. Composite stamps, such as composite stamps disclosedin Ser. No. 12/177,963, filed Aug. 29, 2008, hereby incorporated byreference, may be particularly useful for pickup and release/printsystems, wherein the stamp can be first laminated or contacted with adonor substrate to pickup microstructures or nanostructures from thatdonor substrate and subsequently brought into contact with a receivingsubstrate to which it transfers the microstructures or nanostructures.

“Composite stamp” refers to a stamp having more than one component, suchas more than one material. In some embodiments, a composite stamp ismade from a deformable layer and a rigid support layer, wherein thedeformable and support layers have different chemical compositions andmechanical properties. The deformable layer optionally comprises acomposite polymer layer, such as a reinforcement layer having acombination of one or more polymer and a fiber, such as a glass orelastomeric fiber, particulate, such as nanoparticles or microparticlesor any combinations thereof.

The deformable layer may be an elastomer layer. “Elastomer” or“elastomeric” refers to a polymeric material which can be stretched ordeformed and return to its original shape without substantial permanentdeformation. Elastomers commonly undergo substantially elasticdeformations. Exemplary elastomers useful in the present invention maycomprise, polymers, copolymers, composite materials or mixtures ofpolymers and copolymers. Elastomeric layer refers to a layer comprisingat least one elastomer. Elastomeric layers may also include dopants andother non-elastomeric materials. Elastomers useful in the presentinvention may include, but are not limited to, silicon containingpolymers such as polysiloxanes including poly(dimethyl siloxane) (i.e.PDMS and h-PDMS), poly(methyl siloxane), partially alkylated poly(methylsiloxane), poly(alkyl methyl siloxane) and poly(phenyl methyl siloxane),silicon modified elastomers, thermoplastic elastomers, styrenicmaterials, olefenic materials, polyolefin, polyurethane thermoplasticelastomers, polyamides, synthetic rubbers, polyisobutylene,poly(styrene-butadiene-styrene), polyurethanes, polychloroprene andsilicones.

“Supported” refers to a semiconductor element, such as a micro ornanostructure that will form a semiconductor, that has been adhered tothe stamp's surface (e.g., transfer surface), such that the element iscapable of being transferred to another surface (e.g., a receivingsurface). “Inking” refers to the step of pickup or transfer of micro ornanostructures from a donor substrate to the stamp.

As used herein the expressions “semiconductor component,” “semiconductorelement,” and “semiconductor structure” are used synonymously andbroadly refer to a semiconductor material, structure, device and/orcomponent of a device. Semiconductor elements include high qualitysingle crystalline and polycrystalline semiconductors, semiconductormaterials fabricated via high temperature processing, dopedsemiconductor materials, organic and inorganic semiconductors andcomposite semiconductor materials and structures having one or moreadditional semiconductor components and/or non-semiconductor components,such as dielectric layers or materials and/or conducting layers ormaterials. Semiconductor elements include semiconductor devices anddevice components including, but not limited to, transistors,photovoltaics including solar cells, diodes, light emitting diodes,lasers, p-n junctions, photodiodes, integrated circuits, and sensors. Inaddition, semiconductor elements refer to a part or portion that formsan end functional semiconductor.

“Relief features” refer to protrusions, extensions or projections on anexternal surface of respective posts protruding from a stamp, where eachpost is configured for contact with an individual semiconductor element.The relief features, also referred to herein as a three-dimensionalrelief pattern, facilitate dry-transfer printing of semiconductorelements from a donor substrate to a target or receiver substrate.“Printable surface area” or “printable surface region” refers to thatportion of the stamp used to transfer structures from a donor substrateto a target substrate. A “pattern of relief features” refers to aplurality of features, including a plurality of nanostructures ormicrostructures, such as an array of features. In some embodiments, oneor more relief features may have a different geometry, differentdimension(s) such as height, length or width, or may be made from amaterial resulting in, for example, a different physical parameter suchas an effective Young's modulus for that population.

“Lamination” refers to the process of bonding layers of a compositematerial or a process of producing contact between a first material orlayer and a second layer or material (e.g., such as between the rigidbacking and reinforcement layer, rigid backing and deformable layer,reinforcement layer and deformable layer, and/or semiconductor elementand transfer surface or receiving surface, for example). “Delamination”refers to the stamp transfer surface-semiconductor element separation orthe stamp transfer surface-receiving substrate separation. Inparticular, for embodiments where the stamp has printing posts includingprotruding relief features that are inked with semiconductor elements,delamination rate refers to separation of the printing post surfaceincluding the relief features from the semiconductor elements.Delamination rate may refer to a single post surface delaminating froman individual semiconductor element. Alternatively, delamination ratemay refer to a spatially-averaged rate for all post surfaces within theprintable surface region. In general, processes provided hereinfacilitate high transfer yield and placement accuracy for delaminationrates that are substantially higher than other techniques.

“Substantially similar” refers to a variable that varies less than about10% compared to an average value. For example, a substantially similardelamination rate refers to a rate that varies less than 10% from anaverage rate over the delamination cycle.

“Substrate” refers to a structure or material on which, or in which, aprocess is conducted, such as patterning, assembly and/or integration ofsemiconductor elements. Substrates include, but are not limited to: (i)a structure upon which semiconductor elements are fabricated, deposited,transferred or supported; (ii) a device substrate, for example anelectronic device substrate; (iii) a donor substrate having elements,such as semiconductor elements, for subsequent transfer, assembly orintegration; and (iv) a target substrate for receiving printablestructures, such as semiconductor elements.

“Placement accuracy” refers to the ability of a pattern transfer methodor device to generate a pattern in a selected region of a substrate.“Good placement” accuracy refers to methods and devices capable ofgenerating patterning in a select region of a substrate with spatialdeviations from the absolutely correct orientation less than or equal to5 microns, particularly for generating patterns of semiconductorelements on target substrates.

“Operably connected” refers to a configuration of layers and/or devicecomponents of composite patterning devices of the present invention suchthat functionality of the components or layers are preserved whenconnected. Operably connected layers or device components, refers to anarrangement wherein a force applied to a layer or device component istransmitted to another layer or device component. Operably connectedlayers or device components may be in contact, such as layers havinginternal and/or external surfaces in physical contact. Alternatively,operably connected layers or device components may be connected by oneor more intervening connecting layers, such as thin metal layers orreinforcement layers, positioned between the internal and/or externalsurfaces of two layers or device components, or that run between two ormore layers or components.

The invention may be further understood by the following non-limitingexamples.

Embodiments of the present invention are directed to reversible controlof adhesion, which is a feature employed in systems such as climbingrobots, medical tapes, and stamps for transfer printing. Experimentaland theoretical studies of pressure modulated adhesion between flat,stiff objects and elastomeric surfaces with protruding or sharp featuresof surface relief in improved or optimized geometries are presentedherein. In particular embodiments described in greater detail below, thestrength of non-specific adhesion can be switched by more than threeorders of magnitude, from strong to weak, in a reversible fashion.Implementing these concepts in stamps for transfer printing enablesversatile modes for deterministic assembly of solid materials in micro-or nano-structured forms. Examples described below with reference toprinted two- and three-dimensional collections of silicon platelets andmembranes illustrate some applications of embodiments of the presentinvention. A type of transistor that incorporates a printed gateelectrode, an air gap dielectric, and an aligned array of single walledcarbon nanotubes is described below as an example device that may befabricated in accordance with some embodiments.

Some embodiments of the present invention may arise from observation ofmodes of adhesion in insects and small animals, such as geckos. Inparticular, some such creatures exhibit the ability to adhere to a widevariety of surfaces, to rapidly and reversibly change adhesion strengthbetween strong and weak modes, and to self-clean contaminants. Many ofthese creatures have micro and nanoscale structures with varying levelsof complexity on foot or toe-pads. For example, when adhering to orreleasing from a smooth surface, aphid adhesion organs (pulvilli) areeverted by increased blood pressure or withdrawn by contraction oftibial muscles, respectively. This pressure driven mechanical sagging orretraction of the pulvilli enlarges or diminishes the contacting areas,in a reversible fashion that induces corresponding changes in adhesionstrength. Some embodiments of the present invention provide syntheticmaterials that provide advantages similar to such biological strategiesto yield dry adhesives that offer, for example, switchability inadhesion through changes in thermal and/or mechanical conditions, withone area of possible use in methods for deterministic assembly ofmicro/nanomaterials by transfer printing.

In particular, embodiments of the present invention provide a switchableadhesive surface in which pressure induced sagging of a microstructuredor “textured” elastomeric surface provides reversible levels ofswitchability in non-specific, generalized adhesion. Strong- toweak-adhesion ratios may be higher than 1000 in some embodiments. Thedesigns, which are referred to herein as microtip transfer surfaces, arerobust, reusable and can be cleaned with commercial pressure sensitiveadhesives like Scotch™ tape. These mechanisms of adhesion areincorporated in stamps for printing-based assembly of siliconnanomembranes and platelets on a variety of surfaces, in two and threedimensional layouts that would be difficult or impossible to accommodateusing other methods. As a device example, printing in accordance withembodiments of the present invention is used to form a type of carbonnanotube transistor that uses a nanoscale air gap as a gate dielectric.

The adhesives described herein have potential uses in many applications.In particular, embodiments of the present invention provide advancedcapabilities in the manipulation of stiff, solid micro- or nano-scaleobjects via their selective transfer from one substrate (i.e. donorsubstrate) to another substrate (i.e. receiver substrate) using soft,elastomeric stamps. This transfer printing process enables massivelyparallel assembly of diverse materials (i.e. Si, GaN, GaAs, mica,graphene, silica, and others) in various structural forms (i.e. wires,membranes, plates, with dimensions from a few nanometers to macroscopicscales), with throughputs that correspond to millions of objects perhour. A growing number of applications in micro and nanotechnology maybenefit from or may be enabled by embodiments of the present invention.

The yields in transfer depend on the ability to switch from strong toweak adhesion for retrieval (i.e. ‘inking’) and delivery (i.e.‘printing’), respectively. To increase or maximize the versatility,printing is accomplished using stamps without specialized surfacechemistries or adhesives. While kinetic approaches that exploitviscoelastic effects in the stamps may be useful, the low contrast inadhesion switching (i.e. approximately 3) may limit their broad utility.The experimental results and associated theoretical models describedherein provide alternative design strategies, with enhanced capabilitiesfor printing based assembly, as well as for other areas of use.

FIGS. 1A to 1F illustrate operations and apparatus for transfer printingfrom a donor substrate 101 to a receiver substrate 150 using a stamp 100having an elastomeric, microtip adhesive surface in accordance with someembodiments of the present invention. Referring now to FIG. 1A, anelastomeric stamp 100 includes a plurality of posts 105 protrudingtherefrom. The surface area of the posts 105 define a printable surfacearea of the stamp 100, and each of the posts 105 is configured toretrieve or ink a transferable component 120. Each post 105 includes anelastomeric microtip transfer surface 110 having a plurality of relieffeatures (or “microtips”) 111 protruding therefrom. For example, thegeometry may include four features of pyramidal relief 111 on thesurface 110 of square posts 105. However, other shapes, such ashemispherical shapes, may be employed for the relief features 111 insome embodiments.

The relief features 111 are positioned in a square array placed on anapproximately 1 mm thick backing layer of the same material, configuredto allow mechanical deformation or collapse of the region 115 betweenthe relief features 111 of each post 105 when subjected to a sufficientapplied pressure or force per unit area. The design illustrated in FIG.1A allows for high levels of switching in adhesion, involving complexinteraction between the pressure-controlled contact area and aspects ofsoft adhesion that may be inherent in the viscoelastic nature of theelastomer, as discussed in greater detail herein.

FIG. 1B illustrates retrieval (or “inking”) of the transferablecomponent 120. As shown in FIG. 1B, during retrieval, the stamp 100 isbrought into contact with or pressed against the component 120 with adownward force or pressure that is sufficient to compress the microtips111 and mechanically deform or collapse the region 115 between themicrotips 111 to contact the component 120, thereby increasing ormaximizing the contact area (and as a result, the strength ofgeneralized adhesion, which is typically dominated by van der Waalsinteractions) between the component 120 to be transferred and the stamp100. To provide sufficiently low strengths of adhesion to the donorsubstrate 101, the stamp 100 is retracted from the donor substrate 101at relatively high speeds to retrieve the component 120 in a way thatsimultaneously increases or maximizes its adhesion to the stamp 100through viscoelastic effects, as shown in FIG. 1C.

FIG. 1D illustrates that, immediately after retraction, elasticrestoring forces bring the region 115 of the post 105 back to itsoriginal geometry, such that contact with the component 120 occurs onlyat the uncompressed or “sharp” points of the microtips 111. Then, asshown in FIG. 1E, the stamp 100, inked in this manner, is brought intocontact with or pressed against a receiving surface 150 with a force orpressure that is insufficient to mechanically deform the portion 115between the microtips 111, such that bottom surface of the component 120comes into complete contact with the receiving substrate 150, but theregion 115 between the microtips 111 of the stamp 100 does not collapseto contact the component 120. The stamp 100 is then retracted from thereceiving surface 150 at a relatively low speed to reduce or minimizethe adhesion strength associated with viscoelastic effects, therebyfacilitating delamination or release of the component 120 from the stamp100 to complete the transfer printing assembly process, as shown in FIG.1F.

Precision translation and rotational stages control the positions of thestamp 100 during the various steps in the printing process of FIGS.1A-1F. After each complete sequence of printing, the resultingstructures (e.g., the components 120 on the receiver substrate 150) maybe annealed at about 200° C. to 900° C. (depending on the material ofthe receiver substrate 150) in air for about 3 minutes to reduce oreliminate residual photoresist, and to increase the strength ofadhesion.

The microtip surfaces may be formed with the elastomerpolydimethylsiloxane (PDMS), using casting and curing procedures of softlithography with appropriate templates. PDMS is a transparent elastomerhaving attractive properties such as linear elastic response toelongations of 100% or more, high physical toughness, and excellentfatigue characteristics. FIG. 1G illustrates a PDMS stamp 100 in greaterdetail. In particular, the stamp 100 includes layer of PDMS including aplurality of elastomeric square posts 105 protruding therefrom. Therightmost area of FIG. 1G shows an enlarged view of a post 105, whichincludes an elastomeric microtip adhesive surface 110 having fourpyramidal microtips 111 arranged in a square array. The dimensions shownin FIG. 1G are discussed in detail in the examples that follow.

As shown in FIG. 1H, the fabrication of the microtips 111 involvescasting and curing a layer 100′ of the elastomer polydimethylsiloxane(PDMS, 5:1 mixture of base to curing agent) against a Si (100) wafer 195with a pattern of photolithograpically defined epoxy 185 (100 μm thick).An array of pyramidal pits 190 (15×15 μm squares, 10.6 μm deep,separated by 70 μm with square packing arrangement) is formed in thewafer 195 by anisotropic etching with KOH. The epoxy layer 185 providessquare openings 180 (100×100 μm) with corners aligned to array of pits190. Casting the prepolymer to PDMS 100′ (base oligomer and crosslinkingagent) against the functionalized (trichlorosilane) surface of thiswafer 195, thermally curing the PDMS 100′ (70° C. for >1 hour), and thenpeeling it back yields the stamp 100 including the elastomeric surfaceswith microtips 111.

For purposes of demonstration, transfer printing in accordance with someembodiments may be performed using platelets of silicon (havingdimensions of 100×100 μm; thicknesses of 260 nm or 3 μm) as transferablecomponents. FIG. 1I is a schematic illustration of a process forfabricating silicon platelets in printable configurations by removingthe buried oxide layer from a silicon-on-insulator (SOI) wafer, startingwith SOI wafers 1000 having 3 μm or 260 nm thick top Si layers. Inparticular, as shown in FIG. 1I, the platelets 120′ are defined bypatterning a layer of photoresist (1.5 μm thick) in a square geometry(100 μm×100 μm, square packing arrangement, 300 μm separation), and thenetching the exposed top Si by SF₆ reactive ion etching. Next, wetetching with HF through a mask of photoresist was used to remove theburied oxide 1001 everywhere except for 110×110 μm squares co-centeredwith the squares of silicon. A final pattern of photoresist was used todefined mechanical anchor features 1002 (15×45 μm rectangles, 1.5 μmthick) to tether the silicon squares 120′ to the underlying wafer 1000at each of their four sides, and the remaining oxide 1001 was removed byundercut etching with HF.

FIGS. 2A-2C are scanning electron microscope (SEM) images of a poststructure 105 having a four-tipped layout, as used in the embodiment ofFIGS. 1A-1F. In particular, FIG. 2A illustrates an elastomeric post 105including four microtips 111 on a transfer surface 110, where eachmicrotip 111 is positioned at a corner of the post 105. FIG. 2Billustrates a retrieval or “inking” operation in which the post 105 isbrought into contact with or pressed against a transferable siliconcomponent or platelet 220 (3 μm thick; 100 μm×100 μm area). As shown inFIG. 2B, the pressure or force with which the post 105 is pressedagainst the platelet 220 is sufficient to mechanically deform both themicrotips 111 and a central portion 115 of the post between themicrotips 111 to contact the platelet 220, thereby increasing thecontact area with the platelet 220. FIG. 2C illustrates that, whenpressure is no longer applied and the post 105 is retracted, the centralportion 115 of the post 105 returns to its original shape due to elasticrestoring forces, such that only the sharp points of the microtips 111remain in contact with the platelet 220. The right frames of FIGS. 2Band 2C provide magnified views of one of the microtips 111 during inkingand retraction, respectively, while the lower frames of FIGS. 2B and 2Cillustrate the inking and retraction operations, respectively, usingfinite element modeling.

FIG. 2D is a schematic illustration of the post structure 105 of FIGS.1A-1F. In particular, FIG. 2D illustrates the width (w_(stamp)) of eachpost 105 and the width (w_(microtip)) and height (h_(microtip)) of eachmicrotip 111 relative to a retrieved platelet 220. FIG. 2D alsoillustrates a contact radius (R_(contact)) which is a function of themicrotip cone angle θ (e.g., the angle between opposing sides of eachtip 111) and defines a contact area between each microtip 111 and theretrieved platelet 220. The above stamp dimensions and theirrelationship to adhesion of the platelet 220 to the post 105 of a stamp100 is discussed in greater detail below.

FIGS. 2E and 2F are SEM images showing a post structure 205 having afive-tipped layout according to further embodiments of the presentinvention. In particular, FIG. 2E illustrates a stamp having anelastomeric post 205 including four microtips 211 and a larger microtip215 on a transfer surface 210. Each microtip 211 is positioned at acorner of the post 205, while the larger microtip 215 is positioned at acentral portion of the post 205 between the smaller microtips 211. Asshown in FIG. 2F, after retrieving a platelet 220′ and retracting thepost 205 (including the platelet 220′ thereon) from a donor substrate,the central microtip 215 returns to its original shape due to elasticrestoring forces, and the tip thereof remains in contact with theplatelet 220′. However, one or more of the microtips 211 may no longerbe in contact with the platelet 220′. As such, in the embodiments ofFIGS. 2E and 2F, the platelet 220′ may remain in contact only with thelargest, central microtip 215 in the final stages of the transferprinting process.

FIGS. 2G and 2H illustrate operations and apparatus for transferprinting from a donor substrate 201 to a receiver substrate 250 usingthe five-tipped stamp 200 of FIGS. 2E-2F, for a relatively thickcomponent 220 a (having a thickness of about 3 μm; shown at left side ofFIG. 2G) or a relatively thin component 220 b (having a thickness ofabout 260 nm; shown at right side of FIG. 2G). FIG. 2H illustratesoptical microscope top view images of the stamp, which were collected byviewing through a transparent stamp during various stages of theprinting. As shown in FIGS. 2G and 2H, the stamp 200 is pressed againstthe component 220 a/220 b on the donor substrate 201 with a forcesufficient to collapse the central region of the post 205 including thelargest microtip 215 to contact the component 220 a/220 b. The stamp 200including the component 220 a/220 b adhered thereto is rapidly retractedfrom the donor substrate 201. For the relatively thick component 220 a,the largest microtip 215, located in the central region of the post 205between the smaller microtips 211, is the only point of contact betweenthe stamp 200 and the component 220 a. However, the relatively thincomponent 220 b may partially deform during the retrieval process, suchthat all five of the microtips 211 and 215 contact the component 220 b.The stamp 200 including the component 220 a/220 b adhered thereto isthen pressed against the receiver substrate 250 with a force that isinsufficient to substantially deform the central region of the post 205including the larger microtip 215 thereon, and is then retracted fromthe receiver substrate 250 at a relatively slow speed (in comparison toretraction from the donor substrate 201) such that the component 220a/220 b is delaminated from the stamp 200 and printed onto the receiversubstrate 250.

FIGS. 3A and 3B illustrate force-time and force-distance curvesassociated with contact of a microtip surface with a silicon wafer intransfer printing operations. In particular, the plots of FIGS. 3A and3B illustrate operation for retrieval and delivery, respectively, for asingle post 105 having the four-tipped design (as shown above in FIGS.1A-1F) collected at an approach speed of 5 μm/s, terminated at aspecified load for 5 s, and then retracted at 1 mm/s. In FIGS. 3A and3B, time is indicated on the bottom axes, and distance indicated on thetop axes.

Referring now to FIG. 3A, the maximum tensile force during retractiondefines the strength of adhesion (i.e. pull-off). FIG. 3A shows data fora representative case of full mechanical deformation or collapse of theregion 115 between the microtips 111 under a preload force of about 1 mNduring the retrieval operation, with a retraction speed of about 1 mm/s.Two slopes are evident in the approaching curve 301 a, indicating anincrease in stiffness when the region 115 between the microtips 111collapses and contacts the substrate. The slope in the first regiondefines an effective spring constant associated with compression of themicrotips 111, with a minor contribution from deformation of the post105. The second region includes the elasticity of the post 105 itself,and its elastomeric support. The sharp, negative feature in the curvecollected during retraction 303 a corresponds to rapid release from thecontacting surface, and its magnitude defines the adhesion force (i.e.pull-off).

FIG. 3B summarizes the corresponding cases of approaching 301 b,relaxation 302 b, and retracting 303 b during the delivery operationwithout collapse of the region 115 between the microtips 111, at apreload force of about 0.2 milliNewtons (mN) and a retraction speed ofabout 1 mm/s. In FIG. 3B, the adhesion force is too small to measurewith the load cell. Images collected with an inverted optical microscopeand an SEM (as shown in FIGS. 2B and 2C) suggest effective contact areasin the collapsed and uncollapsed states that correspond to approximately80% and approximately 0.07%, respectively, of the area of the post 105and the microtips 111. The ratio of these areas suggests an expecteddifference in adhesion between the collapsed and uncollapsed states ofmore than about 1000 times; however, as this estimate does not includeviscoelastic effects, it may underestimate the actual difference thatcan be achieved.

FIGS. 3C and 3D illustrate force-speed plots for removing a post havinga microtip surface (FIG. 3C) and a post having a flat surface (FIG. 3D)from the silicon wafer, as a function retraction speed for threedifferent preload cases. The three preload cases with forces of 0.2 mN,1.5 mN, and 3 mN simulate the steps of retrieval (1.5 mN, 3 mN) anddelivery (0.2 mN) in a printing process. Modeling results for themicrotip surface are indicated as a line 320 in FIG. 3C. As shown inFIG. 3C, with preload forces of 1.5 mN and 3 mN (both of which aresufficient to induce mechanical collapse or “sagging” of the centralregion 115 of the post 105), the adhesion force depends strongly onretraction speed. As shown in FIG. 3D, this functional dependence isalso evident in data for the corresponding flat surfaces, and arisesfrom the viscoelastic nature of the PDMS. Significant changes in suchadhesion behaviors was not observed even on repeated cycling tests.Accordingly, embodiments of the present invention employ these combinedgeometric and material effects to provide high levels of switching inadhesion, for use in transfer printing, without the need for surfacechemistries or separate adhesives to guide transfer.

A custom measurement system, as shown by way of example in FIG. 3E, maybe used to quantify the adhesion. An example measurement system includesmotorized x, y stages 399 and a manual tilting stage 398 that supports aprecision load cell 397. Microtip surfaces, such as the microtips 111included on the surface 110 illustrated in FIGS. 1A-1G, are mounted onan independent vertical stage 396 that allows contact with a targetsubstrate (for example, a silicon wafer 395 for the results discussedherein) at controlled speeds and forces. FIG. 3F is a graph illustratingpull-off force data for a stamp with a four-tipped layout in accordancewith embodiments of the present invention as compared to as stamp withflat surface, measured repeatedly with 200 μm/s retraction speed and 2mN preload constantly up to 100 times.

FIGS. 4A-4F are SEM images illustrating example transfer printingresults using thick (about 3 μm) and thin (about 260 nm) siliconplatelets (100×100 μm squares) on different surfaces and in freestanding and multilayer stacked geometries, as achieved withmicrostructured elastomeric surfaces and methods in accordance with someembodiments of the present invention. In particular, FIG. 4A shows suchplatelets printed onto array of square islands (7×7 μm squares,separated by 13 μm with square packing arrangement). The ability totransfer the platelets at high yields without adhesives, particularly onstructured surfaces where contact areas with the receiving surface aremuch smaller than the areas of the platelets themselves, illustrates theutility of microtip designs according to embodiments of the presentinvention, as such capabilities may not be possible with methods thatuse only viscoelastic effects for control.

FIG. 4B illustrates a transfer printing example in accordance withembodiments of the present invention where the platelets are printedonto the rough surface of a film of ultrananocrystalline diamond (2 μmthick, root mean square (rms) roughness>70 nm with sharp facet edges) ona silicon wafer. In FIG. 4B, the contact area is estimated to be lessthan about 1% of the platelet area.

FIGS. 4C and 4D illustrate an example of transfer printing in accordancewith embodiments of the present invention where the printed platelets(with thicknesses of about 3 μm (FIG. 4C) and about 260 nm (FIG. 4D))span the gaps between pairs of silicon bars on receiver substrates.Stamps with five microtips (e.g. four tips 211 at corners of the post205 and central tip 215 therebetween, as shown for example in FIGS. 2Eand 2F) were used for printing the 260 nm thick platelets in FIG. 4D, aslevels of adhesion in the printing mode can be relatively small due tocontact only at the central post in the final stages of release. Thus,transfer printing operations using microstructured elastomeric surfacesaccording to embodiments of the present invention may allow forprecision printing to yield freely suspended structures.

The high yield and versatility of transfer printing methods employingmicrostructured elastomeric surfaces in accordance with embodiments ofthe present invention also enable the formation of complex, threedimensional assemblies, as shown for example in FIGS. 4E and 4F. Inparticular, FIG. 4E illustrates a multilayer configuration of 3 μm thicksilicon platelets printed in a single stack with small incrementalrotations and translations on a flat silicon wafer substrate. FIG. 4Flikewise illustrates the printing of 3 μm thick silicon platelets infour stacks with translational and rotational increments, capped with apair of platelets in the center. These examples demonstrate thattransfer printing using microstructured elastomeric surfaces inaccordance with embodiments of the present invention are capable ofconstructing 3D microstructures and nanostructures that approach thoseconstructed with macro-scale fabrication methods based on assembly ofbuilding blocks, e.g. LEGO® with silicon.

FIG. 5 is a schematic illustration of a device capable of printingsemiconductor elements according to some embodiments of the presentinvention. As shown in FIG. 5, a deformable elastomeric stamp, such asthe stamp 100 described above with reference to FIGS. 1A-1F, isconnected to a print tool head 501. Further detail of tool head 501 isprovided in U.S. patent application Ser. No. 12/177,963, which is herebyincorporated by reference herein. As discussed above, the stamp 100includes a plurality of protruding posts 105, and each of the postsincludes a plurality of three-dimensional relief features 111 protrudingfrom a surface 110 that faces the component to be transferred.Controller 530 is configured to operate the tool head 501 during inkingand printing operations as illustrated above in FIGS. 1A-1F, and asdescribed below with reference to the flowchart of FIG. 6.

FIG. 6 is a flowchart illustrating example operations for transferprinting using stamps with microstructured elastomeric surfaces inaccordance with embodiments of the present invention. As shown in FIG.6, a stamp including an elastomeric post having three-dimensional relieffeatures protruding from a surface thereof is pressed against acomponent on a donor substrate with a first pressure (block 605). Thefirst pressure is sufficient to mechanically deform a region of the postbetween the relief features (as well as the relief features themselves)to contact the component over a first contact area. The stamp isretracted from the donor substrate at a relatively high speed such thatthe component is adhered to the stamp (block 610), thereby retrievingthe component from the donor substrate.

Still referring to FIG. 6, during retraction from the donor substrate,elastic restoring forces return the relief features and the region ofthe post between the relief features to their original states, such thatthe component is adhered to the stamp over a second contact area definedby the ends or tips of the relief features. As noted above, the adhesivestrength of the first contact area may be greater than that of thesecond contact area by a factor of 1000 or more in some embodiments.However, the elastic restoring forces associated with the post are notsufficient to delaminate the component from the stamp. The stampincluding the component adhered thereto is pressed against a receivingsubstrate with a second pressure that is less than the first pressure(block 615). The second pressure is insufficient to mechanically deformthe region of the post between the relief features to contact thecomponent. In particular, the second pressure is sufficient to compressthe relief features without collapse of the region of the posttherebetween, such that the relief features contact the component over athird contact area that is smaller than the first contact area but islarger than the second contact area. The stamp is then retracted fromthe receiving substrate at a relatively low speed to delaminate thecomponent from the stamp (block 620), thereby delivering or “printing”the component onto the receiving substrate.

Accordingly microtip designs as provided by embodiments of the presentinvention can be used in the fabrication of active devices forapplications in areas ranging from photonics and metamaterials tophotovoltaics and electronics.

For example, FIGS. 7A-7C illustrate a transistor 700 that combines aprinted gate electrode 720, an air gap dielectric 710, and a parallelarray of single walled carbon nanotubes 715, which may be fabricatedusing microtip designs as described herein. In particular, FIG. 7Aillustrates SEM images of a carbon nanotube field effect transistor(CNFET) 700 with 100 nm thick air gap dielectric 710 fabricated usingtransfer printing of a heavily doped (p=0.0014 Ωcm) silicon platelet 720(3 μm thick; 100 μm, m×100 μm). The right frames of FIG. 7A providemagnified views of the single walled carbon nanotubes (SWNTs) 715 underthe silicon platelet 720 and 100 nm air gap 710 between the printedplatelet 705 (which provides the gate electrode) and a quartz substrate750 in the left frames. FIG. 7B is a schematic illustration of aperspective view of the gate 720, air gap 710, SWNTs 715, source 725,drain 730, and quartz substrate 750 of the CNFET 700 with dimensionsshown. FIG. 7C is a graph illustrating transfer characteristics for theCNFET 700. Further discussion of the CNFET 700 is provided withreference to the examples below.

FIGS. 8A-8D are graphs illustrating microtip characteristics for use ininking and printing operations according to embodiments of the presentinvention. In particular, FIG. 8A illustrates the contact radius (atzero preload) between the microtips and platelet versus the microtipradius of curvature for the microtip cone angle θ=90°. The asymptote forvanishing microtip radius gives minimal contact radius. FIG. 8Billustrates the minimum height of microtips versus the work of adhesion(normalized by the post width and plane-strain modulus of the stamp) forθ=90°, together with the experimental data for delamination andcollapse. FIG. 8C illustrates the maximum height of microtips versus thepreload for several values of work of adhesion. FIG. 8D illustrates themaximum height of microtips versus the preload, together with theexperimental data for retrieval and failure.

FIGS. 9A and 9B are graphs illustrating forces acting on microstructuredelastomeric surfaces according to embodiments of the present invention.In particular, FIG. 9A illustrates restoring forces associated withcompression of microtips on the surface of an elastomer as a function ofpreload, with corresponding images of finite element modeling results.FIG. 9B is a master plot showing the force required to separate anelastomer surface from a flat substrate, as a function of retractionspeed for the different preload cases shown in FIGS. 3C and 3D. The dataillustrated in FIG. 9B compares the cases of elastomeric posts thatterminate in flat surfaces and elastomeric posts including sets of fourmicrotips, scaled to account for the mechanics of the microtips,according to theoretical modeling.

FIGS. 10A and 10B are graphs illustrating load vs. displacement formicrostructured elastomeric surfaces in accordance with embodiments ofthe present invention according to finite element modeling, as comparedwith experimental data and a contact mechanics model. In particular,FIG. 10A illustrates load-displacement comparison between FEM resultsand experimental data, while FIG. 10B illustrates load-displacementcomparison between FEM results and contact mechanics model for a PDMSmicrotip pressed against a silicon substrate.

Embodiments of the present invention are described in greater detailbelow with reference to the following examples and experimental data.

Pyramidal Relief Feature Example

Referring again to the microtip design of FIGS. 1A-1F, using a pyramidgeometry for each microtip 111 has certain advantages. For example, thepyramid geometry is relatively easy to fabricate in a controlled,lithographic manner using techniques of anisotropic etching in silicon.Also, the radius of curvature of the tip can be extremely small, and isdecoupled from the overall height of the relief feature (i.e. thepyramid). The underlying mechanics principles described herein are notlimited to the pyramid geometry, and thus relief features in accordancewith embodiments of the present invention can also employ other shapes.For instance, embodiments of the present invention may include microtipshaving a conical or hemispherical geometry.

The underlying physics of adhesion in surfaces described herein may berevealed by data analysis guided by theoretical mechanics modeling,which may and identify parameters for improvement or optimization. Forexample, the strength of the low adhesion state, where the adhesionenergy per unit area in the limit of zero peel rate multiplied by thearea of contact at the ends of the microtips, may be considered. FIGS.2A and 2C show a representative case, where the stamp, made of PDMS,incorporates anisotropically etched pits in silicon (100), to yieldmicrotips with radii of curvature, R_(microtip), of less thanapproximately 100 nm. Contact with the silicon causes the microtips todeform, to maintain equilibrium between attraction from surface adhesionand elastic repulsion. These deformations lead to contact areas that areconsiderably larger than those that might be inferred based only on thegeometry of the stamp. Classical models of contact mechanics can beadapted to give analytically the contact radius R_(contact):

$\begin{matrix}{{\frac{R_{contact}}{\gamma/\overset{\_}{E}} = {s\left( {\frac{R_{microtip}}{\gamma/\overset{\_}{E}},\theta} \right)}},} & (1)\end{matrix}$

where s is a non-dimensional function of the microtip cone angle θ(shown in FIG. 2D) and R_(microtip). Finite element analysis yieldssimilar results, as shown in FIG. 2C. The value of R_(contact) scaleslinearly with the work of adhesion γ between the PDMS and the contactingsurface, and inversely with the plane-strain modulus Ē=E/(1−ν²) of PDMS(E-Young's modulus, ν≈0.5—Poisson's ratio). Analysis shows thatR_(contact) decreases with R_(microtip), but reaches an asymptotic valuefor R_(microtip)→0 given by:

$\begin{matrix}{R_{contact}^{\min} = {\frac{32\gamma}{\pi \; \overset{\_}{E}}\tan^{2}{\frac{\theta}{2}.}}} & (2)\end{matrix}$

These analytical models assume symmetric deformations, without anybending or buckling. The SEM and FEM results illustrated in FIGS. 2A-2Dsupport the validity of this assumption.

For the case of a PDMS stamp and a silicon surface, where E=1.8 MPa (31)and γ=155 mJ/m², R_(contact) is approximately the same as R_(contact)^(min) when R_(microtip) is less than approximately 100 nm. When θ=90°between two opposite edges of pyramid (w_(microtip)=√{square root over(2)}h_(microtip) in experiments, FIG. 1A), R_(contact) ^(min) isapproximately 680 nm, which is comparable to 750 nm (evident from FIG.2C), and 732 nm given by the finite element method. The conclusion,then, is that existing methods for producing elastomer surfaces in amaterial like PDMS can achieve values of R_(microtip) well below thevalue which may be needed to realize minimal contact area.

Additional microtip layouts according to embodiments of the presentinvention can further reduce the contact area below that provided by thefour tip design. FIGS. 2E and 2F show examples of such a layoutincluding 5 microtips. As shown in FIG. 2G, during release, separationoccurs first at the microtips at the corners, followed by the microtipat the center, thereby reducing the contact area immediately beforecomplete release to about four times lower than that associated withFIG. 2C. Further reductions may be possible by increasing E (forexample, by using other silicones), or decreasing γ (for example, byusing related elastomers such as perfluoropolyethers), or decreasing themicrotip cone angle θ.

The heights of the microtips and their nearest neighbor separationsrepresent other parameters which may be relevant to the inking andprinting operations described herein. In particular, microtip designsaccording to embodiments of the present invention enable unstablecollapse, with near full area contact (e.g., a contact area that issubstantially similar to or even greater than a cross-sectional area ofthe post or the surface area of a post having a substantially flatsurface) in the compressed state. For a given separation, there may be aminimum height of the microtip, h_(min), below which the elasticrestoring force is too small to bring the relief back to its originalgeometry after pressure induced collapse. This minimum height can bedetermined by equating the strain energy in the compressed PDMS andmicrotips to the adhesion energy between the contacting surfaces, whichgives, for the four tip design,

$\begin{matrix}{{h_{\min} = \sqrt{\frac{w_{stamp}\gamma}{\overset{\_}{E}}\left\lbrack {{3.04\mspace{14mu} {\ln\left( \frac{w_{stamp}\overset{\_}{E}}{\gamma \; \tan^{2}\frac{\theta}{2}} \right)}} - 11.5} \right\rbrack}},} & (3)\end{matrix}$

where w_(stamp) is the width of the post of the stamp. For w_(stamp)=100μm (E=1.8 MPa, γ=155 mJ/m² and θ=90°), the above expression givesh_(min)=8.44 μm, which agrees well with the minimum height ofapproximately 8.5 μm observed systematically in experiments (see FIG.8B). There also exists a maximum height, h_(max), above which theelastic restoring force associated with compression of the microtips isso large that the stamp may rapidly delaminate from the platelet afterthe pull-off force is applied, thereby preventing large contact areasfor removal of the silicon platelet (or other component 120, asillustrated in FIGS. 1A-1F) from a donor substrate. The value of h_(max)can be determined analytically by equating the energy release rate tothe work of adhesion between the PDMS stamp and the silicon platelet.The result is:

$\begin{matrix}{{h_{\max} = {w_{stamp}{f\left( {\frac{P}{w_{stamp}^{2}\overset{\_}{E}},\frac{w_{microtip}}{w_{stamp}},\frac{\gamma}{w_{stamp}\overset{\_}{E}}} \right)}}},} & (4)\end{matrix}$

where f is a non-dimensional function of the applied force P, themicrotip width (w_(microtip), FIG. 1G), and the work of adhesion γ. Foran applied force of 1 mN and w_(microtip)=√{square root over(2)}h_(microtip) (other material and geometry parameters the same asbefore), the maximum height is h_(max)=13.3 μm, which agrees reasonablywell with the maximum height of approximately 12.7 μm observed fromexperimental data (see FIGS. 8C and 8D). These minimum and maximumvalues elucidate criteria that define three possible energy states ofthe stamp and the platelet: (i) platelet retrieval with reliefcollapsed, (ii) platelet retrieval with relief delaminated, (iii)failure in platelet retrieval. The microtip sizes were adjusted oroptimized to obtain the second state for representative preload forces(>1 mN) and retrieval velocities (>200 μm/s). Experiments with differentmicrotips show that when h is approximately 20% smaller or larger thanthis value, states (i) (platelet retrieval with relief collapsed) or(iii) (failure in platelet retrieval) with the same preloads andvelocities could be achieved, respectively. The rate dependence observedin the microtip structures according to embodiments of the presentinvention is similar, to within experimental uncertainties, to that inflat post stamps.

For operation in the retrieval mode, the stamp should be retractedsufficiently quickly that the fracture of the interface between theplatelets and their donor substrate occurs before the viscoelasticfracture of the stamp/platelet interface. During fast retraction, thecompressed microtips may not have time to relax back to their originalshapes; their heights remain small and the overall contact area remainshigh, such that the energy release rate is lower than the work ofadhesion. An analytical viscoelastic model, with creep compliance datafor PDMS, gives a relaxation time of about 0.052 seconds (for pullingspeed of about 460 μm/s), at which the collapsed stamp starts to debondfrom the substrate. For relatively fast retraction (pulling speed>200μm/s), this timescale is roughly consistent with experimentalobservation because the time for complete separation of thestamp/substrate interface is about the same as the time for initialdebonding. Additionally, this viscoelastic analysis predicts a pull-offforce that is in quantitative agreement with the experiments at pullingspeeds>200 μm/s, as shown in FIG. 3C. For pulling speeds<200 μm/s, theanalysis gives a larger pull-off force than the experiments because thedebonding may gradually propagate along the stamp/substrate interfacedue to slow retraction, but the model does not account for crackpropagation along the interface. Improved analysis and comparison toexperiment may require accurate measurements of creep compliance in ourPDMS and direct visualization of the interface using high speed imagingtechniques.

The force-distance curves and other behaviors of FIGS. 3A-3D can also becaptured by mechanics modeling. For the case of FIG. 3A, modelingpredicts two slopes, as observed in experiment: k_(microtip) whencontact occurs only at the microtips, and k_(post) for contact at boththe microtips and the intervening regions. In particular, analysisyields

$\begin{matrix}{\frac{1}{k_{microtip}} = {\frac{1}{k_{post}} + {\frac{1}{{Ew}_{microtip}}{\quad{\left\lbrack {\frac{3\; h_{microtip}}{4\; w_{microtip}} + {\frac{1}{\pi}\left( {\frac{w_{microtip}}{w_{stamp}} - {2\frac{w_{microtip}^{3}}{w_{stamp}^{3}}}} \right){\ln \left( {\frac{w_{stamp}^{2}}{2\; w_{microtip}^{2}} - 1} \right)}}} \right\rbrack.}}}}} & (5)\end{matrix}$

For k_(microtip)=30 N/m and k_(post)=90 N/m shown in FIG. 3A, andh_(microtip)=10.6 μm and width w_(microtip)=√{square root over(2)}h_(microtip)=15 μm from experiments, the left- and right-hand sidesof Eq. (5) give 0.033 m/N and 0.036 m/N, respectively. Thissubstantially similar level of agreement validates the modeling, and itsfurther use in examining the differences between FIG. 3C and FIG. 3D togain insights into the adhesion mechanics. In the collapsed state, themicrotips provide forces that add to the externally applied force neededto cause delamination. This effect can be explored through calculation.In particular, the mechanics models described previously yieldanalytical forms for the restoring force, F, associated with thecompressed microtips. The result takes the form:

$\begin{matrix}{{F = {w_{stamp}^{2}\overset{\_}{E}{g\left( {\frac{P}{w_{stamp}^{2}\overset{\_}{E}},\frac{w_{microtip}}{w_{stamp}},\frac{h_{microtip}}{w_{stamp}}} \right)}}},} & (6)\end{matrix}$

where g is a non-dimensional function of the applied force P, microtipwidth w_(microtip) and height h_(microtip). This force, as shown in FIG.9A, is the same as the preload when only the microtips contact theplatelet. The sudden increase in the restoring force corresponds to thecollapse of post between microtips. This force then increases linearlywith the preload (shown as “post contact” in FIG. 9A), but with reducedslope due the elasticity of the post. This dependence is followed by anonlinear increase, at a reduced rate because the contact area alsoincreases (“zipping of interface” in FIG. 9A). For an applied force of1.5 mN, the total restoring force is 0.63 mN for the collective effectof four microtips with height h_(microtip)=10.6 μm and widthw_(microtip)=√{square root over (2)}h_(microtip)=15 μm (other materialand geometry parameters are the same as above). This restoring force islarger than the preload of about 0.39 mN sufficient to collapse theregions between the microtips (i.e., the position in the curve of FIG.3A that occurs at the point where the linear slope changes) because themicrotips continue to be compressed after the intervening regionscollapse (see FIG. 9A). FIG. 9B illustrates a master plot obtained byshifting the data of FIG. 3D downward along the y-axis by an amountequal to the total restoring force evaluated by modeling, and theplotting results together with the data of FIG. 3C. The overlap of theresulting curves, to within experimental uncertainty, supports themodeling and the associated interpretation of the underlying physics.

Device Assembly Example

To demonstrate a device assembly example, a class of transistor wasbuilt that combines a printed gate electrode 720, an air gap dielectric710, and a parallel array of single walled carbon nanotubes 715. FIG. 7Ashows such a device, with a 100 nm thick air gap dielectric 710 and agate electrode 720 that consists of a heavily doped (ρ=0.0014 Ωcm)silicon platelet (3 μm thick; 100 μm×100 μm), delivered to the devicestructure by printing. Strategically located patterns 740 of thin metalfilms provide support structures at the corners of the platelet 720 todefine its physical separation from the nanotubes 715 (i.e. thethickness of the air gap dielectric 710). After growing aligned singlewalled carbon nanotubes (SWNTs) 715 by chemical vapor deposition (CVD),source and drain electrodes 725 and 730 were defined by electron-beamevaporation with 1 nm thick Ti and 49 nm thick Pd followed byconsecutive lift off process in acetone. A peripheral area of SWNT wasremoved with oxygen reactive ion etching to yield electrically isolateddevices. An air gap spacer 710 with 100 nm thick gold was made byelectron-beam evaporation and lift off process. A heavily doped(p=0.0014 Ωcm) silicon platelet 720 was transfer printed on a patternedPDMS surface, and then the residue of photoresist anchors was removedwith acetone. Finally, the cleaned platelet 720 was transfer printedfrom the PDMS surface to the air gap spacer 710 and annealing processwas performed on a hot plate at 200° C. for 30 minutes in ambient argon.

The variation in source/drain current (I_(DS)) as a function of gatevoltage (V_(GS)) at a source/drain bias (V_(DS)) of −0.05 V for arepresentative device shown in FIG. 7B with a channel length of 5 μm anda channel width of 30 μm is illustrated in FIG. 7C. Gate leakagecurrents less than 10 pA were observed at V_(GS) up to 7 V, where thefield strength is somewhat larger than 2 MV/cm. Increased currents occurfor higher voltages, somewhat lower than those expected based on airbreakdown according to Pashen's law. Accurate models of the capacitancecoupling of the printed gate 720 to the array of tubes 715 (havingdensities of approximately 0.5 tubes/μm) can be used together with themeasured properties shown in FIG. 7C to estimate the mobility to beapproximately 1500 cm²/Vsec, which may be comparable to values reportedfor devices with conventional layouts. The type of device 700 discussedherein may be used in numerous applications, such as sensing in gases orliquids, where both gate modulation and physical access to the nanotubesmay be required.

Contact Radius at Zero Preload

The shape of microtips can be represented by a spherical portion nearthe tip and a conical portion in the cylindrical coordinates (r,z),

$\begin{matrix}{z = {{f(r)} = \left\{ {\begin{matrix}{R_{microtip} - \sqrt{R_{microtip}^{2} - r^{2}}} & {0 \leq r \leq {R_{microtip}\cos \frac{\theta}{2}}} \\{\frac{r}{\tan \frac{\theta}{2}} - {R_{microtip}\left( {\frac{1}{\sin \frac{\theta}{2}} - 1} \right)}} & {r > {R_{microtip}\cos \frac{\theta}{2}}}\end{matrix}.} \right.}} & ({S1})\end{matrix}$

The contact mechanics model relates the radius of contact R_(contact) tothe above shape function f(r), work of adhesion γ, and plane-strainmodulus Ē by

$\begin{matrix}{{{{\frac{\overset{\_}{E}R_{contact}}{2\pi}\left\lbrack {\frac{\delta}{R_{contact}} - {\int_{0}^{R_{contact}}\frac{{f^{\prime}(r)}\; {r}}{\sqrt{R_{contact}^{2} - r^{2}}}}} \right\rbrack}^{2} = \gamma},}\ } & ({S2})\end{matrix}$

where δ is related to the preload P by

$\begin{matrix}{P = {2\; \overset{\_}{E}{\int_{0}^{R_{contact}}{\left\lbrack {\delta - {t{\int_{0}^{t}\frac{{f^{\prime}(r)}{r}}{\sqrt{t^{2} - r^{2}}}}}} \right\rbrack {{t}.}}}}} & ({S3})\end{matrix}$

For zero preload P=0, δ is given by

$\begin{matrix}{\delta = {\frac{1}{R_{contact}}{\int_{0}^{R_{contact}}{\sqrt{R_{contact}^{2} - r^{2}}{f^{\prime}(r)}\ {{r}.}}}}} & ({S4})\end{matrix}$

Its substitution in Eq. (S2) gives the equation for R_(contact)

$\begin{matrix}{{\frac{\overset{\_}{E}}{2\pi \; R_{contact}^{3}}\left\lbrack {\int_{0}^{R_{contact}}\frac{r^{2}{f^{\prime}(r)}{r}}{\sqrt{R_{contact}^{2} - r^{2}}}} \right\rbrack}^{2} = {\gamma.}} & ({S5})\end{matrix}$

For the shape function in Eq. (S1), Eq. (S5) gives the followingequation for the ratio of radii

${\eta = \frac{R_{microtip}}{R_{contact}}},$

$\begin{matrix}{{\eta - \frac{\cos^{- 1}\left( {\eta \; \cos \frac{\theta}{2}} \right)}{\tan \frac{\theta}{2}} - {\eta \frac{\sqrt{1 - {\eta^{2}\cos^{2}\frac{\theta}{2}}}}{\sin \frac{\theta}{2}}} + {\left( {1 + \eta^{2}} \right)\ln \frac{\sqrt{1 - {\eta^{2}\cos^{2}\frac{\theta}{2}}} + {\eta \; \sin \frac{\theta}{2}}}{1 + \eta}} + {2\sqrt{\eta}\sqrt{\frac{2{\pi\gamma}}{\overset{\_}{E}R_{microtip}}}}} = 0.} & ({S6})\end{matrix}$

This gives the implicit expression in Eq. (1). The contact radius,normalized by

$\frac{\gamma}{\overset{\_}{E}},$

is shown in FIG. 8A, and so is R_(contact) for material properties inthe experiment. For R_(microtip)→0, it gives analytically the asymptotein Eq. (2).

Equation (S6) holds for conical contact between the microtips andplatelet for a relatively small microtip radius of curvature,

$\begin{matrix}{\frac{\overset{\_}{E}R_{microtip}}{\gamma} \leq {\frac{8\pi \; \cos^{3}\frac{\theta}{2}}{\left\lbrack {{\left( {1 + {\cos^{2}\frac{\theta}{2}}} \right){\ln \left( {\tan \frac{\theta}{4}} \right)}} + {\cos \frac{\theta}{2}}} \right\rbrack^{2}}.}} & ({S7})\end{matrix}$

For microtip radius of curvature exceeding this value, the contactbetween the microtips and platelet remains in the spherical portion, andthe corresponding contact radius has been obtained analytically. Thecontact radii corresponding to conical and spherical contact are alsoshown graphically in FIG. 8A.

Finite Element Analysis of Contact Radius

The contact radii in Eqs. (1) and (2) are derived from classical modelsof contact mechanics, developed for the case of a rigid indenter incontact with a soft material. Similar models can be applied to softindenters in contact with hard materials. For example, researchers havemeasured the indentation load-displacement curve for a conical indenterof soft rubber in contact with a hard, soda-lime glass. The Young'smoduli of rubber (2.45 MPa) and glass (70 GPa) are comparable to thoseof PDMS (1.8 MPa) and silicon (130 GPa), respectively. Table S1summarizes the geometry and elastic properties of the conical rubberindenter with a round tip.

TABLE S1 Geometry and elastic properties of the rubber indenter ConeMaximum Tip Young's Poisson's angle radius radius modulus ratio 60degree 5 mm 0.23 mm 2.45 MPa 0.4999999

This axisymmetric indentation problem was studied using the finiteelement method (FEM), which accounts for the geometric nonlinearity(large change of indenter shape) during indentation. Axisymmetricelements were used for the rubber indenter, including the detailedgeometry of the indenter tip. The element size was approximately 0.0345mm, which is 7 times smaller than the indenter tip radius, and 150 timessmaller than the maximum indenter radius. Refined meshes were used toensure that the numerical results converge. The contact between therubber indenter and the glass expands from an initial cone tip to aconical region as the indentation load increases. A finite sliding, hardcontact model was used, to allow for the possibility of sliding betweencontact surfaces without interpenetration. The normal and shear stresswere continuous within the contact process zone. The friction at thecontact interface was also accounted for, but it had negligible effecton the indentation load-displacement: the difference betweenfrictionless contact and contact with a large friction coefficient wasless than 0.2%. The results of the indentation load versus displacement(shown in FIG. 10A) indicate excellent agreement between FEM andexperiments. This outcome validates the use of FEM for a soft indenterin contact with a hard material.

TABLE S2 Geometry and elastic properties of PDMS microtips PyramidMaximum Tip Young's Poisson's angle height radius modulus ratio 90degree 10.6 μm 100 nm 1.8 MPa 0.48

Table S2 summarizes the geometry and elastic properties of pyramidmicrotips of PDMS used in the experiments. Silicon served as thecontacting substrate. The element size was approximately 1.5 nm, whichis about 70 times smaller than the indenter tip radius, and about 7,000times smaller than the maximum height of microtip. FIG. 10B shows theresulting force versus displacement curve on each microtip, and acomparison to the contact mechanics model (with cone angle 90°)specified in Eqs. (S2) and (S3), in which P and represent theindentation load and displacement, respectively. The numerical andanalytical results agree well at small displacements, but begin todeviate as the displacement increases beyond a couple of microns. Thepresent use of the contact mechanics model involves the determination ofcontact area in the limit of extremely small displacements, associatedwith zero imposed compressive load. The results in FIGS. 10A and 10Bindicate that the model may be applicable for embodiments of the presentinvention.

A more direct validation of the contact mechanics model is to use FEM todetermine the contact radius for the experimental system. To accomplishthis goal, the microtips were compressed into contact with the silicon,and then the load was released completely, which delaminates themicrotip/platelet interface with a work of adhesion γ=155 mJ/m². FEMgives a contact radius of 732 nm, which is slightly larger than 680 nmobtained from Eq. (2) based on the contact mechanics model. Both values,however, agree (within experimental uncertainties of approximately 100nm) with that determined from analysis of scanning electron microscopeimages (i.e., 750 nm). The lower right frame of FIG. 2C shows thedeformed FEM mesh of final contact (in this zero compressive loadregime).

Minimum Height of Microtips

A minimum height of the microtips may correspond to the state ofvanishing preload at which the elastic energy in the stamp due to thecollapse of the post equals the adhesion energy between the stamp andplatelet. The latter equals the product of work of adhesion γ andcontact area, while the former can be obtained using an approach basedon fracture mechanics, which accounts for the finite geometry of thestamp, such as the stamp width w_(stamp) and contact radius R_(contact)between the microtips and platelet. The contact area may be determinedanalytically by reducing or minimizing the total potential energy, whichequals the elastic energy in the stamp subtracted by the adhesionenergy. The minimum height of microtips may be defined analytically as

$\begin{matrix}{{h_{\min} = \sqrt{\frac{w_{stamp}\gamma}{\overset{\_}{E}}\left\lbrack {{3.04\mspace{14mu} {\ln \left( \frac{w_{stamp}}{R_{contact}} \right)}} - 4.44} \right\rbrack}},} & ({S8})\end{matrix}$

where the factors 3.04 and 4.44 result from the stress intensity factorfor finite geometry in fracture mechanics. The substitution of theasymptote in Eq. (2) for R_(contact) leads to the analytical expressionin Eq. (3). FIG. 8B illustrates that a minimum height of microtips(normalized by post width w_(stamp)) increases with the work of adhesionγ, but decreases with the plane-strain modulus of the stamp. The minimumheight for the material properties and post width in experiments is alsoshown. The experimental data for delamination (above the curve) andcollapse (on or below the curve) agree with the model.

Analysis of Stamp Collapse Process

For microtip heights larger than h_(min) in Eq. (3), the process ofstamp collapse includes 4 stages as the preload P increases, (i)microtip contact, during which only microtips contact the platelet; (ii)post collapse, which corresponds to a sudden increase of contact areabetween the post and platelet; (iii) post contact, during which thecontact area remains the same as the preload increases; and (iv) zippingof interface, which corresponds to the increase of contact area with thepreload.

-   (i) microtip contact: The deformation in the microtips and post is    studied by linear elasticity, where the microtips are subject to    uniaxial compression, and the post is subject to the preload and    reaction forces from the microtips.-   (ii) post collapse: The analysis is similar to that for the minimum    height, except that the total potential energy includes the external    work of the preload. It gives following three equations to determine    the ratio c_(collapse) of contact area to stamp area at collapse,    the corresponding critical load P_(collapse), and the compressed    height h_(collapse) of microtips at collapse,

$\begin{matrix}{{{{\frac{1}{F_{1}(b)}\left( \frac{h_{collapse}}{w_{stamp}} \right)^{2}\frac{K\left( \frac{c_{collapse}}{b} \right)}{K\left\lbrack \sqrt{1 - \left( \frac{c_{collapse}}{b} \right)^{2}} \right\rbrack}} - {\frac{h_{collapse}}{w_{stamp}}{\frac{P_{collapse}}{w_{stamp}^{2}\overset{\_}{E}}\left\lbrack {c_{collapse} + {\left( {\sqrt{{2\; {bc}_{collapse}} + {2\; c_{collapse}^{2}}} - {2\; c_{collapse}}} \right){F_{2}\left( {b - c_{collapse}} \right)}}} \right\rbrack}} + {\frac{3\; h_{microtip}}{4\; w_{stamp}}\left( {1 - b} \right)\left( {\ln \frac{h_{collapse}}{h_{microtip}}} \right)^{2}} - {\frac{4\; h_{microtip}}{3\; w_{stamp}}\frac{1}{1 - b}\left( \frac{P_{collapse}}{w_{stamp}^{2}\overset{\_}{E}} \right)^{2}} - {\frac{2\gamma}{w_{stamp}\overset{\_}{E}}c_{collapse}}} = 0},} & ({S9}) \\\left. {\frac{1}{F_{1}(b)}\left( \frac{h_{collapse}}{w_{stamp}} \right)^{2}\frac{}{c}\left\{ \frac{K\left( \frac{c}{b} \right)}{K\left\lbrack \sqrt{1 - \left( \frac{c}{b} \right)^{2}} \right\rbrack} \right\}} \middle| {}_{c = c_{collapse}}{{- \frac{h_{collapse}}{w_{stamp}}}\frac{P_{collapse}}{w_{stamp}^{2}\overset{\_}{E}}{\quad{{{\begin{bmatrix}{1 + {\left( {\frac{b + {2\; c_{collapse}}}{\sqrt{{2\; {bc}_{collapse}} + {2\; c_{collapse}^{2}}}} - 2} \right){F_{2}\left( {b - c_{collapse}} \right)}} -} \\{\left( {\sqrt{{2\; {bc}_{collapse}} + {2\; c_{collapse}^{2}}} - {2\; c_{collapse}}} \right){F_{2}^{\prime}\left( {b - c_{collapse}} \right)}}\end{bmatrix} - \frac{2\gamma}{w_{stamp}\overset{\_}{E}}} = 0},}}} \right. & ({S10}) \\{{{{\frac{1}{F_{1}(b)}\frac{h_{collapse}^{2}}{w_{stamp}h_{microtip}}\frac{K\left( \frac{c_{collapse}}{b} \right)}{K\left\lbrack \sqrt{1 - \left( \frac{c_{collapse}}{b} \right)^{2}} \right\rbrack}} + {\frac{3\left( {1 - b} \right)}{4}\ln \frac{h_{collapse}}{h_{microtip}}} - {\frac{h_{collapse}}{w_{stamp}}{\frac{P_{collapse}}{w_{stamp}^{2}\overset{\_}{E}}\left\lbrack {{{F_{2}\left( {b - c_{collapse}} \right)}\left( {\sqrt{{2\; {bc}_{collapse}} + {2\; c_{collapse}^{2}}} - {2\; c_{collapse}}} \right)} - 1 + c_{collapse}} \right\rbrack}}} = 0},} & ({S11})\end{matrix}$

where

${b = {1 - \frac{4\; w_{microtip}^{2}}{3\; w_{stamp}^{2}}}},{{K(k)} = {\int_{0}^{\pi/2}\ \frac{\phi}{\sqrt{1 - {k^{2}\sin^{2}\phi}}}}}$

is the elliptical function of the first kind, F₁(k)=−0.417−1.07 ln(1−k),and F₂(k)=(1−0.25 k+0.093 k²−0.005 k³)/√{square root over (1−0.5 k)}.

-   (iii) post contact: The contact area is the same as that in (ii),    but the energy release rate at the boundary of contact decreases as    the preload increases. The compressed height h′ of microtips    decreases with the increase of preload, and is given by

$\begin{matrix}{{{{\frac{1}{F_{1}(b)}\frac{h^{\prime 2}}{w_{stamp}h_{microtip}}\frac{K\left( \frac{c_{collapse}}{b} \right)}{K\left\lbrack \sqrt{1 - \left( \frac{c_{collapse}}{b} \right)^{2}} \right\rbrack}} + {\frac{3\left( {1 - b} \right)}{4}\ln \frac{h^{\prime}}{h_{microtip}}} - {\frac{h^{\prime}}{h_{microtip}}{\frac{P}{w_{stamp}^{2}\overset{\_}{E}}\left\lbrack {{{F_{2}\left( {b - c_{collapse}} \right)}\left( {\sqrt{{2\; {bc}_{collapse}} + {2\; c_{collapse}^{2}}} - {2\; c_{collapse}}} \right)} - 1 + c_{collapse}} \right\rbrack}}} = 0},} & ({S12})\end{matrix}$

-   (iv) zipping of interface: The energy release rate at the boundary    of contact reaches and remains at zero, and the contact area    increases with the preload. The ratio c of contact area to stamp    area increases with the preload, while the opposite holds for the    compressed height h′ of microtips, and they are given by

$\begin{matrix}{{{{\frac{1}{F_{1}(b)}\frac{h^{\prime 2}}{w_{stamp}h_{microtip}}\frac{K\left( \frac{c}{b} \right)}{K\left\lbrack \sqrt{1 - \left( \frac{c}{b} \right)^{2}} \right\rbrack}} + {\frac{3\left( {1 - b} \right)}{4}\ln \frac{h^{\prime}}{h_{microtip}}} - {\frac{h^{\prime}}{h_{microtip}}{\frac{P}{w_{stamp}^{2}\overset{\_}{E}}\left\lbrack {{{F_{2}\left( {b - c} \right)}\left( {\sqrt{{2\; {bc}} + {2\; c^{2}}} - {2\; c}} \right)} - 1 + c} \right\rbrack}}} = 0},} & ({S13}) \\{h^{\prime} = {{F_{1}(b)}\frac{P}{w_{stamp}\overset{\_}{E}}\left( {1 - \frac{c}{b}} \right)\sqrt{\frac{c}{2}\left( {b + c} \right)}{K\left\lbrack \sqrt{1 - \left( \frac{c}{b} \right)^{2}} \right\rbrack}{{F_{2}\left( {b - c} \right)}.}}} & ({S14})\end{matrix}$

This analysis provides the slope change in the preload-distance curve(shown FIG. 3A), maximum height of the microtips, and restoring force inthe microtips, as follows.

Slope Change in the Preload-Distance Curve

The distance in FIG. 3A before post collapse includes the (compressive)displacements in microtips and in the post. The microtips are subject touniaxial compression, while the post is modeled as a semi-infinite solidsubject to remote compression and forces from the microtips on thesurface. The ratio of preload to this distance gives the slopek_(microtip)

$\begin{matrix}{{\frac{1}{k_{microtip}} = {{\frac{1}{w_{microtip}E}\left\lbrack {\frac{3\; h_{microtip}}{4\; w_{microtip}} + {\frac{1}{\pi}\left( {\frac{w_{microtip}}{w_{stamp}} - \frac{2\; w_{microtip}^{2}}{w_{stamp}^{3}}} \right){\ln \left( {\frac{w_{stamp}^{2}}{2\; w_{microtip}^{2}} - 1} \right)}}} \right\rbrack} + \frac{h_{stamp}}{w_{stamp}^{2}E}}},} & ({S15})\end{matrix}$

where h_(stamp) is the effective height of the stamp (FIG. 1G). Thechange of distance in FIG. 3A after post collapse also includescontributions from the microtips and from the post, but the former maybecome negligible as compared to the latter. The ratio of preloadincrement to distance increment gives the slope k_(post)

$\begin{matrix}{\frac{1}{k_{post}} = {\frac{h_{stamp}}{w_{stamp}^{2}E}.}} & ({S16})\end{matrix}$

Eqs. (S15) and (S16) lead to Eq. (5).

Restoring Force in Microtips

The restoring force microtips is given by

$\begin{matrix}{{F = {{P\left\lbrack {{\left( {\sqrt{{2\; {bc}} + {2\; c^{2}}} - {2\; c}} \right){F_{2}\left( {b - c} \right)}} - 1 + c} \right\rbrack} - {\frac{w_{stamp}\overset{\_}{E}}{F_{1}(b)}\frac{K\left( \frac{c}{b} \right)}{K\left( \sqrt{1 - \frac{c^{2}}{b^{2}}} \right)}h^{\prime}}}},} & ({S17})\end{matrix}$

where c and h′ are determined from Eqs. (S9) to (S14) for stages(ii)-(iv).

Maximum Height of Microtips

A maximum height of microtips may be determined by equating the energyrelease rate to the work of adhesion, which gives the following relationto determine c′

$\begin{matrix}{{\frac{\pi \; P^{2}}{4\; w_{stamp}^{3}\overset{\_}{E}}{\left( {b - c} \right)\left\lbrack {F_{2}\left( {b - c} \right)} \right\rbrack}^{2}} = {\gamma.}} & ({S18})\end{matrix}$

Eq. (S14) then gives explicitly h′. The maximum height of microtips,h_(max), is obtained from Eq. (S13) by replacing h_(microtip) withh_(max). FIG. 8C shows that the maximum height of microtips (normalizedby post width w_(stamp)) increases with the preload as well as the workof adhesion γ, but decreases with the plane-strain modulus of the stamp.The maximum height for the material properties and post width inexperiments is also shown. The experimental data for retrieval (belowthe curve) and failure (above the curve) agree with the model.

Viscoelastic Analysis

For operation in retrieval mode, the PDMS stamp is retractedsufficiently quickly to ensure that the platelet/substrate interfacefractures, but the stamp/platelet interface does not, due to effects ofviscoelastic behavior in the PDMS. The creep compliance of PDMS, is amaterial property that governs this process. This quantity can berepresented by a piece-wise relation

$\begin{matrix}{\frac{C(t)}{C(\infty)} = \left\{ {\begin{matrix}{0.198 \times \left\lbrack {6.14 + {\log (t)}} \right\rbrack} & {0.0001 < t < 0.08} \\1 & {0.08 < t}\end{matrix},} \right.} & ({S19})\end{matrix}$

which is a non-decreasing function of time t (unit: second). Theviscoelastic energy release rate G is related to the stress intensityfactor K(t) via the creep compliance by

$\begin{matrix}{{G = {\frac{1}{2\; \overset{\_}{E}}\frac{C(t)}{C(\infty)}{K^{2}(t)}}},} & ({S20})\end{matrix}$

where Ē is the plane-strain modulus of PDMS, and the factor ½ accountsfor the elastic mismatch between PDMS and silicon. The stress intensityfactor K(t) is given by

$\begin{matrix}{{K_{I} = {{\frac{\overset{\_}{E}h^{''}}{F_{1}(b)}\sqrt{\frac{\pi}{2\; w_{stamp}{c\left\lbrack {1 - \left( \frac{c}{b} \right)^{2}} \right\rbrack}}}\frac{1}{K\left\lbrack \sqrt{1 - \left( \frac{c}{b} \right)^{2}} \right\rbrack}} + {\frac{P^{''}}{2\; w_{stamp}}\sqrt{\frac{\pi \left( {b - c} \right)}{w_{stamp}}}{F_{2}\left( {b - c} \right)}}}},} & ({S21})\end{matrix}$

where w_(stamp) is the stamp width, b and functions F₁, F₂ and K aredefined after Eq. (S11), c is solved from Eqs. (S13) and (S14), P″ isthe pull-off force, and the microtip height h″ is related to P″ by

$\begin{matrix}{{{{\frac{1}{F_{1}(b)}\frac{h^{''2}}{w_{stamp}h_{microtip}}\frac{K\left( \frac{c}{b} \right)}{K\left\lbrack \sqrt{1 - \left( \frac{c}{b} \right)^{2}} \right\rbrack}} + {\frac{3\left( {1 - b} \right)}{4}\ln \frac{h^{''}}{h_{microtip}}} + {\frac{h^{''}}{h_{microtip}}{\frac{P^{''}}{w_{stamp}^{2}\overset{\_}{E}}\left\lbrack {{{F_{2}\left( {b - c} \right)}\left( {\sqrt{{2\; {bc}} + {2\; c^{2}}} - {2\; c}} \right)} - 1 + c} \right\rbrack}}} = 0},} & ({S22})\end{matrix}$

which is identical to Eq. (S13) except that P and h′ are replaced by −P″and h″, respectively.

The pull-off force is related to the pulling speed v_(pulling) and timet by

P″=w _(stamp) E(ν_(pulling) t−L _(compression)),  (S23)

after the compression force P is relaxed, where L_(compression) is thecompressed distance of the stamp due to P, and L_(compression)=20 μmfrom FIG. 3A.

The stamp/platelet interface will not delaminate if the viscoelasticenergy release rate remains smaller than the work of adhesion γ, i.e.,

G<γ.  (S24)

For the given material properties, the creep compliance in Eq. (S19),and a pulling speed v_(pulling)=460 μm/s, the above inequality gives atime of about 0.052 second for the stamp/platelet interface stating todebond. The pull-off force is then obtained from Eq. (S23).

Accordingly, embodiments of the present invention provide methods fordeterministic assembly of solid microscale or nanoscale parts into twoand three dimensional configurations, and some theoretical foundationfor understanding key design parameters. The embodiments discussedherein provide experimental data and theoretical models on the use ofmicrostructures of relief on elastomeric surfaces to achieve pressureinduced switching in adhesion strength. Theoretically guided designimprovement or optimization yields high levels of control, with morethan three orders of magnitude difference between the forces measured instrong and weak adhesive states. These characteristics enable transferprinting reliably and repeatedly with very high yield (almost 100%) innew modes, for numerous applications. Adhesion may be further increasedrelative to the corresponding flat surface using vacuum effects ornotched features on the sidewalls of the posts. These and otherstructural designs can be further enhanced through the introduction ofnew materials, using guidance from mechanical models similar to thosepresented herein and the viscoelastic effect model on elastomericmicrotip surface adhesion during high speed part retrieval.

Various embodiments are described herein with reference to flowchartillustrations of computer-implemented methods, apparatus (systems and/ordevices) and/or computer program products. It is understood that a blockof the flowchart illustrations, and combinations of blocks in theflowchart illustrations, can be implemented by computer programinstructions that are performed by one or more computer circuits. Thesecomputer program instructions may be provided to a processor circuit ofa general purpose computer circuit, special purpose computer circuit,and/or other programmable data processing circuit to produce a machine,such that the instructions, which execute via the processor of thecomputer and/or other programmable data processing apparatus, transformand control transistors, values stored in memory locations, and otherhardware components within such circuitry to implement thefunctions/acts specified in the block diagrams and/or flowchart block orblocks, and thereby create means (functionality) and/or structure forimplementing the functions/acts specified in the block diagrams and/orflowchart block(s)

These computer program instructions may also be stored in a tangible,non-transitory computer-readable medium that can direct a computer orother programmable data processing apparatus to function in a particularmanner, such that the instructions stored in the computer-readablemedium produce an article of manufacture including instructions whichimplement the functions/acts specified in the flowchart blocks.

The computer program instructions may also be loaded onto a computerand/or other programmable data processing apparatus to cause a series ofoperational steps to be performed on the computer and/or otherprogrammable apparatus to produce a computer-implemented process suchthat the instructions which execute on the computer or otherprogrammable apparatus provide steps for implementing the functions/actsspecified in the flowchart blocks.

It should also be noted that in some alternate implementations, thefunctions/acts noted in the blocks may occur out of the order noted inthe flowcharts. For example, two blocks shown in succession may in factbe executed substantially concurrently or the blocks may sometimes beexecuted in the reverse order, depending upon the functionality/actsinvolved. Moreover, the functionality of a given block of the flowchartsmay be separated into multiple blocks and/or the functionality of two ormore blocks of the flowcharts and/or block diagrams may be at leastpartially integrated. Finally, other blocks may be added/insertedbetween the blocks that are illustrated.

All references throughout this application, for example patent documentsincluding issued or granted patents or equivalents; patent applicationpublications; and non-patent literature documents or other sourcematerial; are hereby incorporated by reference herein in theirentireties, as though individually incorporated by reference, to theextent each reference is at least partially not inconsistent with thedisclosure in this application (for example, a reference that ispartially inconsistent is incorporated by reference except for thepartially inconsistent portion of the reference).

The terms and expressions which have been employed herein are used asterms of description and not of limitation, and there is no intention inthe use of such terms and expressions of excluding any equivalents ofthe features shown and described or portions thereof, but it isrecognized that various modifications are possible within the scope ofthe invention claimed. Thus, it should be understood that although thepresent invention has been specifically disclosed by preferredembodiments, exemplary embodiments and optional features, modificationand variation of the concepts herein disclosed may be resorted to bythose skilled in the art, and that such modifications and variations areconsidered to be within the scope of this invention as defined by theappended claims. The specific embodiments provided herein are examplesof useful embodiments of the present invention and it will be apparentto one skilled in the art that the present invention may be carried outusing a large number of variations of the devices, device components,methods steps set forth in the present description. As will be obviousto one of skill in the art, methods and devices useful for the presentmethods can include a large number of optional composition andprocessing elements and steps.

It will be understood that when an element such as a layer, region orsubstrate is referred to as being “on” or extending “onto” anotherelement, it can be directly on or extend directly onto the other elementor intervening elements may also be present. In contrast, when anelement is referred to as being “directly on” or extending “directlyonto” another element, there are no intervening elements present. Itwill also be understood that when an element is referred to as being“connected” or “coupled” to another element, it can be directlyconnected or coupled to the other element or intervening elements may bepresent. In contrast, when an element is referred to as being “directlyconnected” or “directly coupled” to another element, there are nointervening elements present.

It will also be understood that, although the terms first, second, etc.may be used herein to describe various elements, these elements shouldnot be limited by these terms. These terms are only used to distinguishone element from another. For example, a first element could be termed asecond element, and, similarly, a second element could be termed a firstelement, without departing from the scope of the present invention.

Furthermore, relative terms, such as “lower” or “bottom” and “upper” or“top,” may be used herein to describe one element's relationship toanother element as illustrated in the Figures. It will be understoodthat relative terms are intended to encompass different orientations ofthe device in addition to the orientation depicted in the Figures. Forexample, if the device in one of the figures is turned over, elementsdescribed as being on the “lower” side of other elements would then beoriented on “upper” sides of the other elements. The exemplary term“lower”, can therefore, encompasses both an orientation of “lower” and“upper,” depending of the particular orientation of the figure.Similarly, if the device in one of the figures is turned over, elementsdescribed as “below” or “beneath” other elements would then be oriented“above” the other elements. The exemplary terms “below” or “beneath”can, therefore, encompass both an orientation of above and below.

The terminology used in the description of the invention herein is forthe purpose of describing particular embodiments only and is notintended to be limiting of the invention. As used in the description ofthe invention and the appended claims, the singular forms “a”, “an” and“the” are intended to include the plural forms as well, unless thecontext clearly indicates otherwise. It will also be understood that theterm “and/or” as used herein refers to and encompasses any and allpossible combinations of one or more of the associated listed items. Itwill be further understood that the terms “comprises” and/or“comprising,” when used in this specification, specify the presence ofstated features, integers, steps, operations, elements, and/orcomponents, but do not preclude the presence or addition of one or moreother features, integers, steps, operations, elements, components,and/or groups thereof.

Embodiments of the invention are described herein with reference tocross-section illustrations that are schematic illustrations ofidealized embodiments (and intermediate structures) of the invention. Assuch, variations from the shapes of the illustrations as a result, forexample, of manufacturing techniques and/or tolerances, are to beexpected. Thus, embodiments of the invention should not be construed aslimited to the particular shapes of regions illustrated herein but areto include deviations in shapes that result, for example, frommanufacturing. In other words, the regions illustrated in the figuresare schematic in nature and their shapes are not intended to illustratethe actual shape of a region of a device and are not intended to limitthe scope of the invention.

When a group of substituents is disclosed herein, it is understood thatall individual members of that group and all subgroups, are disclosedseparately. When a Markush group or other grouping is used herein, allindividual members of the group and all combinations and subcombinationspossible of the group are intended to be individually included in thedisclosure.

Every formulation or combination of components described or exemplifiedherein can be used to practice the invention, unless otherwise stated.

Whenever a range is given in the specification, for example, atemperature range, a size or distance range, a time range, or acomposition or concentration range, all intermediate ranges andsubranges, as well as all individual values included in the ranges givenare intended to be included in the disclosure. It will be understoodthat any subranges or individual values in a range or subrange that areincluded in the description herein need not be necessarily included inand/or can be excluded from the claims herein.

All patents and publications mentioned in the specification areindicative of the levels of skill of those skilled in the art to whichthe invention pertains. References cited herein are incorporated byreference herein in their entirety to indicate the state of the art asof their publication or filing date and it is intended that thisinformation can be employed herein, if needed, to exclude specificembodiments that are in the prior art.

All functional equivalents, of any such materials and methods areintended to be included in this invention. The terms and expressionswhich have been employed are used as terms of description and not oflimitation, and there is no intention that in the use of such terms andexpressions of excluding any equivalents of the features shown anddescribed or portions thereof, but it is recognized that variousmodifications are possible within the scope of the invention claimed.Thus, it should be understood that although the present invention hasbeen specifically disclosed by preferred embodiments and optionalfeatures, modification and variation of the concepts herein disclosedmay be resorted to by those skilled in the art, and that suchmodifications and variations are considered to be within the scope ofthis invention as defined by the appended claims.

1. An apparatus for printing transferable components, the apparatuscomprising: a stamp comprising at least one elastomeric post protrudingtherefrom, the post having a surface configured for contact with arespective transferable component and including three-dimensional relieffeatures protruding therefrom; a transfer printing tool head includingthe stamp mounted thereon; and a controller configured to operate thetransfer printing tool head to press the stamp including the postprotruding therefrom against the respective transferable component on adonor substrate with a first pressure sufficient to mechanically deformat least a portion of the relief features and a region of the postbetween the relief features such that the post contacts the respectivetransferable component over a first contact area, retract the stamp fromthe donor substrate such that the respective transferable component isadhered to the stamp, press the stamp including the respectivetransferable component adhered thereto against a receiving substratewith a second pressure that is less than the first pressure such thatthe post contacts the component over a second contact area that issmaller than the first contact area, and retract the stamp from thereceiving substrate to delaminate the respective transferable componentfrom the stamp and print the respective transferable component onto thereceiving substrate.
 2. The apparatus of claim 1, wherein the secondpressure is insufficient to mechanically deform the region of the postbetween the relief features to contact the respective transferablecomponent.
 3. The apparatus of claim 1, wherein the controller isconfigured to operate the transfer printing tool head to press the stampwith the first pressure to contact the relief features of the stamp withthe component on the donor substrate such that the region of the posttherebetween does not contact the component, and then compress therelief features and collapse the region of the post therebetween tocontact the component to define the first contact area.
 4. The apparatusof claim 3, wherein the first contact area is substantially similar to across-sectional area of the post taken along a plane parallel to thesurface thereof.
 5. The apparatus of claim 3, wherein the controller isconfigured to operate the transfer printing tool head to remove thefirst pressure from the stamp during retraction from the donor substrateto substantially reverse compression of the relief features and collapseof the region of the post therebetween such that the respectivetransferable component is adhered to the stamp by ends of one or more ofthe relief features over the second contact area.
 6. The apparatus ofclaim 5, wherein an elastic restoring force of the post responsive toremoval of the first pressure is insufficient to delaminate thecomponent from the stamp.
 7. The apparatus of claim 6, wherein theelastomeric post and the relief features protruding from the surfacethereof comprise polydimethylsiloxane (PDMS), and wherein an adhesivestrength provided by the first contact area is greater than that of thesecond contact area by three or more orders of magnitude.
 8. Theapparatus of claim 1, wherein the controller is configured to operatethe transfer printing tool head to retract the stamp from the donorsubstrate at a first speed to adhere the respective transferablecomponent thereto, and to retract the stamp from the receiving substrateat a second speed that is less than the first speed to delaminate therespective transferable component therefrom.
 9. The apparatus of claim1, wherein the relief features are positioned around a periphery of thesurface of the post.
 10. The apparatus of claim 9, wherein thethree-dimensional relief features comprise first three-dimensionalrelief features, and wherein the surface of the post further includes asecond three-dimensional relief feature that protrudes from the regionthereof between the first relief features.
 11. The method of claim 10,wherein the second three-dimensional relief feature is larger than thefirst three-dimensional relief features in at least one dimension.